American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Finite-time synchronization of competitive neural networks with mixed delays
  • DCDS-B Home
  • This Issue
  • Next Article
    Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux
December  2016, 21(10): 3637-3654. doi: 10.3934/dcdsb.2016114

Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays

Thanh-Anh Nguyen 1, , Dinh-Ke Tran 2, and  Nhu-Quan Nguyen 3,

1. 

Vietnam Education Publishing House, 81 Tran Hung Dao, Hanoi, Vietnam
2. 

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
3. 

Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam

Received  November 2015 Revised  August 2016 Published  November 2016

We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.
Keywords: measure of non-compactness, Weak stability, MNC estimate., fixed point theory, condensing map.
Mathematics Subject Classification: Primary: 35B35, 37C75; Secondary: 47H08, 47H1.
Citation: Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114
References:
[1]

B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal., 72 (2010), 3190-3208. doi: 10.1016/j.na.2009.12.016.

[2]

N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 1601-1622. doi: 10.1002/mma.3172.

[3]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.

[4]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.

[5]

T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Syst. Appl., 10 (2001), 89-116.

[6]

J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23 (2010), 960-965. doi: 10.1016/j.aml.2010.04.016.

[7]

N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations, J. Integral Equations Appl., 26 (2014), 145-170. doi: 10.1216/JIE-2014-26-2-145.

[8]

C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689. doi: 10.1016/j.na.2009.09.007.

[9]

C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett., 22 (2009), 865-870. doi: 10.1016/j.aml.2008.07.013.

[10]

E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., (Supplement) (2007), 277-285.

[11]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453. doi: 10.2307/2160321.

[12]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.

[13]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[14]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[15]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.

[16]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II, Osaka J. Math., 27 (1990), 797-804.

[17]

C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal., 51 (2002), 765-782. doi: 10.1016/S0362-546X(01)00861-6.

[18]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[20]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[21]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.

[22]

F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[24]

R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett., 51 (2000), 492-498. doi: 10.1209/epl/i2000-00364-5.

[25]

J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.

show all references

References:
[1]

B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal., 72 (2010), 3190-3208. doi: 10.1016/j.na.2009.12.016.

[2]

N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 1601-1622. doi: 10.1002/mma.3172.

[3]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.

[4]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.

[5]

T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Syst. Appl., 10 (2001), 89-116.

[6]

J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23 (2010), 960-965. doi: 10.1016/j.aml.2010.04.016.

[7]

N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations, J. Integral Equations Appl., 26 (2014), 145-170. doi: 10.1216/JIE-2014-26-2-145.

[8]

C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689. doi: 10.1016/j.na.2009.09.007.

[9]

C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett., 22 (2009), 865-870. doi: 10.1016/j.aml.2008.07.013.

[10]

E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., (Supplement) (2007), 277-285.

[11]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453. doi: 10.2307/2160321.

[12]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.

[13]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[14]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[15]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.

[16]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II, Osaka J. Math., 27 (1990), 797-804.

[17]

C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal., 51 (2002), 765-782. doi: 10.1016/S0362-546X(01)00861-6.

[18]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[20]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.

[21]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.

[22]

F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[24]

R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett., 51 (2000), 492-498. doi: 10.1209/epl/i2000-00364-5.

[25]

J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.

[1]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[2]

Roberto Livrea, Salvatore A. Marano. A min-max principle for non-differentiable functions with a weak compactness condition. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1019-1029. doi: 10.3934/cpaa.2009.8.1019
[3]

Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[4]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
[5]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[6]

Nicholas Long. Fixed point shifts of inert involutions. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[7]

Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4051-4059. doi: 10.3934/dcds.2022045
[8]

Yu Liu, Ting Zhang. On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021307
[9]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[10]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709
[11]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[12]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[13]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979
[14]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[15]

Alain Bensoussan, Miroslav Bulíček, Jens Frehse. Existence and compactness for weak solutions to Bellman systems with critical growth. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1729-1750. doi: 10.3934/dcdsb.2012.17.1729
[16]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[17]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404
[18]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[19]

Xiangfeng Yang. Stability in measure for uncertain heat equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6533-6540. doi: 10.3934/dcdsb.2019152
[20]

Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2139-2154. doi: 10.3934/cpaa.2021061

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (162)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy